Ergodic and Geometric Theory of
Conservative and Hamiltonian Flows
Departamento de Matem´atica
Faculdade de Ciˆencias da Universidade do Porto 2011
Ergodic and Geometric Theory of
Conservative and Hamiltonian Flows
Tese submetida `a Faculdade de Ciˆencias da Universidade do Porto para a obten¸c˜ao do grau de Doutor em Matem´atica
We learn to do things by doing the things we are learning to do.
Aos meus pais e ao meu marido.
It is a pleasure to thank the many people who made this thesis possible. Firstly, I would like to express my gratitude to my PhD supervisors, Professor Jorge Rocha and Professor M´ario Bessa, whose expertise, under-standing and patience extremely helped in my PhD experience. I appreciate their vast knowledge and skill in many areas, their careful reading and as-sistance in writing papers. Thank you for the invaluable encouragement, motivation, suggestions and excellent guidance during these last years. It was really rewarding to work with them!
I also thank to the Mathematics Department of the Faculty of Sciences of the University of Porto, specially to the professors, secretaries and librarians, which always worked hard to give knowledge and very good work conditions to the students, which are essential to the development of a PhD thesis.
I must also acknowledge the elements of the Center for Applied Math-ematics and Economics (CEMAPRE ) and of the Center of MathMath-ematics and Fundamental Applications (CMAF ), because of their friendly reception, hints and enlightening opinions.
Thank you to the Federal University of Alagoas (UFAL), in Macei´o, and specially to Professor Krerley Oliveira, and to National Institute of Pure and Applied Mathematics (IMPA), in Rio de Janeiro, for the reception and the support given during my stay. These two visits were very productive and unforgettable. I had the opportunity to meet some PhD students, now great
this opportunity.
I am grateful to all my PhD colleagues, for providing a stimulating and fun environment in which to learn and grow. I am especially indebted to
ˆ
Angela Cardoso and Davide Azevedo, for our philosophical debates, ex-changes of knowledge and venting of frustration during the PhD program, which helped to enrich the experience. Thank you because of the friendship, help and encouragement. It was really funny and important for me to be with you these last years!
I recognize that this thesis would not have been possible without the financial assistance of the Funda¸c˜ao para a Ciˆencia e a Tecnologia (schol-arship SFRH/BD/33100/2007) and the partial support of the Center of Mathematics of the University of Porto (CMUP) and of the Funda¸c˜ao para a Ciˆencia e a Tecnologia project PTDC/MAT/099493/2008, and I express my big gratitude to those agencies.
Lastly, and most importantly, a very special thanks goes out to my sweet-heart, S´ergio Oliveira: thank you for the endless love, encouragement and dedication. I love you! I also wish to thank my parents, Maria Judite F. F. Ferreira and Domingos L. Ferreira, and my brother, Nuno D. F. Ferreira, for providing a loving environment for me through my entire life. They supported me, understood me and loved me. To them I dedicate this thesis.
Esta tese cont´em resultados que contribuem para o desenvolvimento da teoria da dinˆamica conservativa e da dinˆamica Hamiltoniana.
Inicialmente consideram-se sistemas dinˆamicos conservativos em tempo cont´ınuo, definidos em variedades Riemannianas, suaves, fechadas e conexas. Neste contexto ´e provada a C1-conjectura da estabilidade estrutural, assim como resultados que
rela-cionam a hiperbolicidade uniforme com as propriedades de sombreamento e de expan-sividade. Por fim, ´e descrito um cen´ario geral para a dinˆamica cont´ınua conservativa de sistemas definidos em variedades com dimens˜ao superior a 3.
Um C1-campo vectorial com divergˆencia nula satisfaz a propriedade estrela se
qual-quer campo vectorial com divergˆencia nula numa C1-vizinhan¸ca do campo inicial tem
todas as singularidades e todas as ´orbitas fechadas hiperb´olicas. Nesta tese prova-se que todo o C1-campo vectorial com divergˆencia nula com a propriedade estrela ´e
uniforme-mente hiperb´olico e, em particular, n˜ao possui singularidades. Segundo este resultado, provar a hiperbolicidade uniforme para C1-campos vectoriais com divergˆencia nula equiv-ale a provar que o campo satisfaz a propriedade estrela. Este resultado ´e posteriormente utilizado para provar que um C1-campo vectorial com divergˆencia nula e estruturalmente
est´avel ´e, de facto, uniformemente hiperb´olico.
Posteriormente, prova-se a equivalˆencia entre as seguintes quatro propriedades: - um C1-campo vectorial com divergˆencia nula est´a no C1-interior do conjunto dos campos vectoriais expansivos com divergˆencia nula;
- um C1-campo vectorial com divergˆencia nula est´a no C1-interior do conjunto dos
campos vectoriais com divergˆencia nula que verificam a propriedade de sombreamento; - um C1-campo vectorial com divergˆencia nula est´a no C1-interior do conjunto dos
campos vectoriais com divergˆencia nula que verificam a propriedade de sombreamento Lipschitz;
- um C1-campo vectorial com divergˆencia nula ´e uniformemente hiperb´olico.
O ingrediente chave nestas provas ´e a caracteriza¸c˜ao dos campos vectoriais com divergˆencia nula com a propriedade estrela como sendo uniformemente hiperb´olicos.
Em [18], Bessa e Rocha descrevem um cen´ario geral para a dinˆamica conservativa em dimens˜ao 3. Nesta tese generaliza-se este resultado para sistemas definidos em variedades com dimens˜ao superior a 3. Prova-se que um C1-campo vectorial com divergˆencia nula
nestas condi¸c˜oes pode ser C1-aproximado por um campo vectorial com divergˆencia nula uniformemente hiperb´olico ou ent˜ao por um C1-campo vectorial com divergˆencia nula com ciclos heterodimensionais.
A ´ultima parte desta tese re´une resultados de dinˆamica Hamiltoniana. Seja H um Hamiltoniano definido numa variedade simpl´ectica M , e ∈ H(M ) ⊂ R e EH,e uma
com-ponente conexa sem singularidades de H−1({e}). Um sistema Hamiltoniano, seja um tripleto (H, e, EH,e), ´e uniformemente hiperb´olico se a componente EH,e´e uniformemente
hiperb´olica. Por outro lado, um sistema Hamiltoniano (H, e, EH,e) ´e um sistema
Hamil-toniano estrela se todas as ´orbitas fechadas em EH,e s˜ao uniformemente hiperb´olicas e
o mesmo vale para uma componente conexa de ˜H−1({˜e}), perto de EH,e, para qualquer
˜
H numa C2-vizinhan¸ca de H e para qualquer ˜e numa vizinhan¸ca de e. Neste contexto, prova-se que um sistema Hamiltoniano estrela definido numa variedade simpl´ectica de dimens˜ao 4 ´e uniformemente hiperb´olico. Prova-se ainda a conjectura da estabilidade estrutural para sistemas Hamiltonianos em variedades de dimens˜ao 4.
Por fim, mostra-se que, dado um Hamiltoniano gen´erico H, existe um conjunto aberto e denso S(H) em H(M ) tal que, para qualquer e ∈ S(H), toda a componente conexa EH,e ⊂ H−1({e}) ´e topologicamente misturadora. O resultado essencial para
concluir esta prova ´e uma vers˜ao do lema da conex˜ao de pseudo-´orbitas para Hamilto-nianos. Nesta tese ´e apresentado o enunciado do lema utilizado, assim como uma ideia da sua prova. Este resultado gen´erico ´e relevante, na medida em que permite obter a prova de resultados como a dicotomia de Newhouse para Hamiltonianos, entre outros. Contudo, estas aplica¸c˜oes s˜ao direccionadas para um trabalho futuro.
This thesis contains results on conservative and on Hamiltonian dynamics.
Here, we include the proof of the C1-structural stability conjecture, as well as re-sults relating uniform hyperbolicity, shadowing and expansiveness properties for C1 -divergence-free vector fields defined on a closed, connected and smooth Riemannian manifold with dimension greater than 2. When the dimension of the manifold is greater than 3, we also describe a general scenario for this kind of dynamics.
A C1-divergence-free vector field satisfies the star property if any divergence-free
vector field in some C1-neighborhood has all singularities and all closed orbits hyperbolic. We prove that any divergence-free vector field satisfying the star property is uniformly hyperbolic. This result is relevant because, from it, to prove uniform hyperbolicity for divergence-free vector fields it is enough to show that the vector field satisfies the star property. Afterwards, this result is used to prove that a C1-structurally stable
divergence-free vector field is, in fact, a uniformly hyperbolic divergence-divergence-free vector field, beyond other results.
Later, we prove that the following properties are equivalent:
- a C1-divergence-free vector field is in the C1-interior of the set of expansive
divergence-free vector fields;
- a C1-divergence-free vector field is in the C1-interior of the set of divergence-free vector fields which satisfy the shadowing property ;
- a C1-divergence-free vector field is in the C1-interior of the set of divergence-free
vector fields which satisfy the Lipschitz shadowing property ; - a C1-divergence-free vector field is uniformly hyperbolic.
Again, a cornerstone to prove this result is the equality between star and uniformly xxi
hyperbolic C1-divergence-free vector-fields, obtained before.
In [18], Bessa and Rocha describe a general scenario for the conservative dynamics in dimension 3. In this thesis, we generalize this result for manifold with dimension greater that 3, by proving that any divergence-free vector field can be C1-approximated by a
uniformly hyperbolic divergence-free vector field, or else by a divergence-free vector field exhibiting a heterodimensional cycle.
Now, let H be a Hamiltonian defined on a symplectic manifold M , e ∈ H(M ) ⊂ R and EH,e a connected component of H−1({e}) without singularities. A Hamiltonian
system, say a triplet (H, e, EH,e), is uniformly hyperbolic if EH,e is uniformly hyperbolic.
A Hamiltonian system (H, e, EH,e) is a Hamiltonian star system if all the closed orbits of
EH,e are hyperbolic and the same holds for a connected component of ˜H−1({˜e}), close
to EH,e, for any ˜H in some C2-neighborhood of H and for any ˜e in some neighborhood
of e. In this context, we show that a Hamiltonian star system defined on a 4-dimensional symplectic manifold is uniformly hyperbolic. Moreover, we prove the structural stability conjecture for Hamiltonian systems defined on a 4-dimensional symplectic manifold.
In the last part of this thesis, we show that, given a C2-generic Hamiltonian H,
there exists an open and dense set S(H) in H(M ) such that, for any e ∈ S(H), every EH,e ⊂ H−1({e}) is topologically mixing. The most important ingredient to show this
result is a version of the connecting lemma for pseudo-orbits of Hamiltonians, whose highlights of the proof are also stated. This theorem is relevant, because it allows us to show results as the Newhouse Dichotomy for Hamiltonians, among others. But these applications are postponed to a future work.
A1
µ(M ) Set of Anosov divergence-free vector fields. 6
A2
ω(M ) Set of Anosov Hamiltonian systems. 10
Crit(X) Set of closed orbits and singularities of the vector field X. 4
e Real scalar, called energy of the Hamiltonian H. 8
EH,e Connected component of H−1({e}), called energy hypersurface. 8
E1
µ(M ) Set of expansive divergence-free vector fields. 15
F C1
µ(M ) Set of far from heterodimensional cycles divergence-free
vector fields. 31
ϕtH(x) Transversal linear Poincar´e flow at the point x. 55
G1
µ(M ) Set of divergence-free star vector fields. 5
G2
ω(M ) Set of star Hamiltonian systems. 9
H Hamiltonian function. 8
HC1
µ(M ) Set of divergence-free vector fields admitting
heterodimensional cycles. 30
KS1
µ(M ) Kupka-Smale’s residual set. 13
LS1
µ(M ) Set of Lipschitz shadowing divergence-free vector fields. 15
O(X) Set of Oseledets points associated to the vector field X. 24
OX(x) Xt-orbit of the point x. 29
Pt
X(x) Linear Poincar´e flow at the point x. 25
P er(X) Set of closed orbits of the vector field X. 4
P erπ(X) Set of closed orbits with period less or equal than π of X. 4
P erπ(X) Set of closed orbits with period greater than π of X. 4
PR1µ(M ) Pugh-Robinson’s residual set. 32
S1
µ(M ) Set of shadowing divergence-free vector fields. 15
Sing(X) Set of singularities of the vector field X. 4
SS1
Acknowledgments xv
Resumo xix
Abstract xxi
Symbols index xxv
1 Introduction and results’ statement 1
1.1 Structural stability conjecture . . . 1 1.2 Shadowing and expansiveness . . . 14 1.3 General scenario for dynamics . . . 17 1.4 Topological transitivity . . . 18
2 Conservative dynamics 23
2.1 Definitions and auxiliary results . . . 23 2.1.1 Lyapunov exponents and classification of closed orbits . . . 23 2.1.2 Linear Poincar´e flow and hyperbolicity . . . 25 2.1.3 Heterodimensional cycles . . . 29 2.1.4 C1-perturbation results . . . 31
2.2 Proof of the conservative results . . . 35 2.2.1 Star property and uniform hyperbolicity . . . 35 2.2.2 Proof of the structural stability conjecture . . . 41 2.2.3 Boundary of A1µ(M ) . . . 43
2.2.4 Shadowing and uniform hyperbolicity . . . 44 2.2.5 Expansiveness and uniform hyperbolicity . . . 46 2.2.6 Heterodimensional cycles and uniform hyperbolicity . . . 48
3 Hamiltonian dynamics 53
3.1 Definitions and auxiliary results . . . 53 3.1.1 Some notes on Hamiltonian dynamics . . . 53 3.1.2 Transversal linear Poincar´e flow and hyperbolicity . . . 55 3.1.3 Topological dimension . . . 58 3.1.4 Homoclinic classes . . . 59 3.1.5 Resonance relations . . . 59 3.1.6 Pseudo-orbits . . . 60 3.1.7 Lift axiom . . . 60 3.1.8 Perturbation flowboxes . . . 61 3.1.9 Covering families . . . 64 3.1.10 Avoidable closed orbits . . . 66 3.1.11 C2-perturbation results . . . 68
3.2 Connecting Lemma for pseudo-orbits . . . 70 3.3 Proof of the Hamiltonian results . . . 74 3.3.1 Openness and structural stability . . . 75 3.3.2 Star property and uniform hyperbolicity . . . 78 3.3.3 Structural stability conjecture . . . 83 3.3.4 Boundary of A2ω(M4) . . . 84 3.3.5 Auxiliary lemmas . . . 85 3.3.6 Energy hypersurfaces as homoclinic classes . . . 88 3.3.7 Generic topological mixing . . . 90
Conclusions and future work 95
Appendix 101
1.1 Representation of a flow. . . 4 1.2 Representation of the Poincar´e first return map. . . 5 1.3 Representation of the sets G1(M ) and Gµ1(M ). . . 6
1.4 Representation of a Hamiltonian function H. . . 8 1.5 Representation of energy hypersurfaces. . . 8 1.6 Representation of a regular energy level. . . 9 1.7 Representation of a analytic continuation of EH,e. . . 9
1.8 Vector field X isolated in the boundary of a set V. . . 12 1.9 Representation of a critical point p of a Kupka-Smale vector field. . . 13 1.10 Representation of a pseudo-orbit. . . 14 1.11 Representation of an expansive vector field’s orbit. . . 16 1.12 Representation of the analytic continuation of EH,e. . . 20
2.1 Representation of the spectrum of a hyperbolic, a parabolic, a completely elliptic and an elliptic closed orbit, respectivelly. . . 25 2.2 Transformation of a completely elliptic closed orbit, with no simple
char-acteristic multipliers, into a hyperbolic closed orbit. . . 25 2.3 Representation of the linear Poincar´e flow. . . 26 2.4 Representation of a heterodimensional cycle. . . 30 2.5 Perturbation given by the Closing Lemma. . . 32 2.6 Representation of a flowbox. . . 33 2.7 Representation of the action of the flow PYτ(p). . . 34
3.1 Spectrum of a symplectomorphism. . . 56 3.2 Representation of a pseudo-orbit on EH,e. . . 60
3.3 Representation of a tiled cube of the chart (U, ϕ). . . 62 3.4 Representation of a pseudo-orbit preserving the tiling. . . 62 3.5 Perturbation in a tiled cube. . . 63 3.6 Representation of a covering family of EH,e. . . 65
3.7 Covering family of EH,e outside V. . . 66
3.8 Representation of an avoidable closed orbit γ. . . 67 3.9 Perturbation given by the Pasting Lemma for Hamiltonians. . . 69 3.10 Perturbation given by the Connecting Lemma for pseudo-orbits. . . 70 3.11 Representation of the stable and unstable cones. . . 75 3.12 Preservation of the volume of a box. . . 82
ONE
INTRODUCTION AND RESULTS’ STATEMENT
This thesis is a contribution to issues concerning on the structural stability conjecture, on the shadowing and expansiveness properties of a dynamical system, on the description of a general scenario for dynamics and on the generic transitivity. These questions will be addressed from the standpoint of conservative and Hamiltonian dynamics.
This chapter brings together the main notation and assumptions in order to properly state the main results.
1.1
Structural stability conjecture
One of the most challenging problems in the modern theory of dynamical systems, posed by Palis and Smale in 1970, is the well-known structural stability conjecture (see [61]).
Conjecture 1.1 A Cr-structurally stable system satisfies the Axiom A and the strong transversality conditions, for r ≥ 1.
Let S be a system defined on a closed manifold. The notion of structural stability was firstly introduced in the mid 1930’s by Andronov and Pontrjagin (see [4]) and this concept is intrinsically related to uniform hyperbolicity.
Roughly speaking, a system is uniformly hyperbolic if the tangent bundle splits into two invariant sub-bundles, one where the action is uniformly contracting and other where the action is uniformly expanding, and, in the continuous-time case, a one dimensional
fiber including the direction of the flow. A system S is Cr-structurally stable (r ≥ 1) if there exists a Cr-neighborhood U of S such that any other system in U is topologically conjugated to S.
We say that the system S satisfies the Axiom A property if the closure of its closed orbits is equal to the non-wandering set, Ω(S), and, moreover, this set is hyperbolic. Notice that a conservative system satisfying the Axiom A property is actually uniformly hyperbolic, since its non-wandering set coincides with the entire manifold. By the spectral decomposition of an Axiom A system S, we have that Ω(S) = ∪k
i=1Λi, where each set
Λi is called a basic piece. We define an order relation by Λi ≺ Λj if there exists x
(outside Λi ∪ Λj) such that α(x) ⊂ Λi and ω(x) ⊂ Λj. The system S has a cycle if
there exists a cycle with respect to ≺ (see [72], for more details).
A cornerstone on the structural stability conjecture was the remarkable proof for C1
-diffeomorphisms, achieved by Ma˜n´e, in [50]. In fact, in the early 1980’s, Ma˜n´e started to define the set F1 as the set of diffeomorphisms having a C1-neighborhood U such
that every diffeomorphism inside U has all periodic orbits of hyperbolic type. A system in F1 is called a star system or a system satisfying the star property. It is known that
Ω-stable diffeomorphisms belong to F1 and that if f ∈ F1 then Ω(f ) = P er(f ) (see [35, 52]). Thus, the structural stability conjecture is contained in the following.
Conjecture 1.2 The non-wandering set of a star system is hyperbolic.
The set F1 is related to the structural stability since the proof that a C1-structural stable system satisfies the Axiom A property mainly uses the fact that the system is in F1. We point out that classic results imply that being in F1 is a necessary condition to
satisfy the Axiom A and the strong transversality conditions (see [50] and the references wherein).
In [51], Ma˜n´e proved Conjecture 1.2 for diffeomorphisms defined on surfaces: any surface diffeomorphism of F1 satisfies the Axiom A and the no-cycle conditions. Later,
in [43], Hayashi extended this result for higher dimensions. In 1988, Ma˜n´e presented a proof of Conjecture 1.1 for C1-diffeomorphisms (see [50]). We point out that, after the proof of the C1-structural stability conjecture for diffeomorphisms, Hayashi proves this conjecture for C1-flows, in [41, 42]. Later Gan gives a different proof of this conjecture
for C1-flows (see [36]). Recently, Bessa and Rocha presented, in [20], a proof of the C1 -structural stability conjecture for conservative flows defined on a 3-dimensional manifold. Nevertheless, the Cr-structural stability conjecture remains wide open for higher
topologies (r ≥ 2). This is explained, in particular, because many of the C1-perturbation
arguments, as the Closing Lemma, the Connecting Lemma and the Franks Lemma, are either unknown or they are false in higher topologies (see further details in [40, 65, 68, 80]).
Even for the continuous-time case, the proof of Conjecture 1.1 is simplified if we firstly prove Conjecture 1.2. In this context, the set analogous to F1 is traditionally
denoted by G1, in which the hyperbolicity of the flow equilibria is also imposed.
The first results on this thesis are about the proof of Conjecture 1.2 for conservative star flows defined on high-dimensional manifolds and also for 4-dimensional Hamiltonian systems. These results will be used later to prove Conjecture 1.1 for high-dimensional conservative flows and for 4-dimensional Hamiltonian flows. In order to properly state these results, let us introduce some definitions.
From now on, Md, sometimes called M , denotes a d-dimensional, (d ≥ 2), compact,
boundary-less, connected and smooth Riemannian manifold, endowed with a volume form, which has associated a measure µ, called the Lebesgue measure. Also, denote by dist the Riemannian distance and consider, for > 0 and p ∈ M , the open balls B(p) = {x ∈ M : dist(x, p) < }.
Denote by Xr(M ) the set of vector fields defined on M , endowed with the Cr
Whitney topology (r ≥ 1). If the divergence of a Cr-vector field X is zero then we call
X a Cr-divergence-free vector field. Let Xrµ(M ) denote the set of divergence-free vector fields defined on M , endowed with the induced Cr Whitney topology. A Cr-vector field X : M → T M generates a flow Xt : M → M , which is a smooth 1-parameter group for t ∈ R, satisfying
d dtX
t|
t=s(p) = X(Xs(p)) and X0 = id.
If X is a divergence-free vector field then Xtis called a conservative flow. The linear part of the flow Xt, called tangent flow, DXpt: TpM −→ TXt(p)M , for p ∈ M , satisfies
Xt(p)
p X(p)
Figure 1.1: Representation of a flow.
the linearized differential equation d
dtDX
t
p = (DXXt(p)) ◦ DXpt,
where DXp : TpM −→ TpM . Let supp(X) = {x ∈ M : X(x) 6= ~0} denote the support
of X. From now on, we are restricted to the C1-topology (r = 1).
A closed orbit γ of X is a non-constant integral curve γ : [a, b] → M of X such that γ(a) = γ(b). We define b as the smallest number greater than a satisfying γ(a) = γ(b). Observe that the period of γ is b − a. For simplicity, sometimes we call p ∈ γ a closed orbit. So, the set of closed orbits associated to the vector field X is denoted by
P er(X) = {p ∈ M : ∃ t > 0 , Xt(p) = p}.
Given a closed orbit γ and any p ∈ γ, if π > 0 is the least number such that Xπ(p) = p then γ is a closed orbit with period π.
Denote by P erπ(X) the set of closed orbits with period less or equal than π of the
vector field X and by P erπ(X) the set of closed orbits with period greater than π of
the vector field X. Obviously, P er(X) = P erπ(X) ∪ P erπ(X).
The set of singularities of the vector field X is denoted by Sing(X) = {p ∈ M : X(p) = ~0}.
Singularities and closed orbits of X are called critical points and are denoted by Crit(X) = Sing(X) ∪ P er(X).
If p /∈ Sing(X) then p is called a regular point and if Sing(X) = ∅ then M is said regular.
Before stating the definition of star vector fields for the continuous-time case, let us explain what does mean a singularity and a closed orbit to be hyperbolic.
Let γ be a closed orbit of X, take p ∈ γ and denote by Σ a (dim(M )−1)-transversal section to X at p. Poincar´e defined a map f from ˜Σ ⊂ Σ to Σ, called the Poincar´e first return map of the trajectories on Σ, such that, for any point x ∈ Σ in a small neighborhood of p, the ω-trajectory of x will intersect Σ again at some point y at some time t close to the period of p.
p Σ ˜ Σ x f (x)
Figure 1.2: Representation of the Poincar´e first return map.
A closed orbit γ of X is hyperbolic if p ∈ γ is a hyperbolic fixed point of the Poincar´e first return map. A singularity q of a C1-vector field X is hyperbolic if the eigenvalues
of DXq are not purely imaginary. We say that any element of Crit(X) is hyperbolic, if
any singularity and any closed orbit of X is hyperbolic.
Definition 1.1 A C1-vector field X is a star vector field if there exists a C1
-neighbor-hood U of X in X1(M ) such that, for any Y ∈ U , any element of the set Crit(Y ) is
hyperbolic. Moreover, a vector field X ∈ X1
µ(M ) is a divergence-free star vector field
if there exists a C1-neighborhood U of X in X1
µ(M ) such that, for any Y ∈ U , any
element of the set Crit(Y ) is hyperbolic. Note that if X ∈ X1µ(M ) is a star vector field then X is a divergence-free star vector field. The set of C1-star vector fields is denoted by G1(M ) and the set of C1-divergence-free star vector fields is denoted by Gµ1(M ).
Observe that, in the previous definition, the hyperbolicity imposed at the critical points is not uniform. So, the hyperbolicity constants depend on the critical point.
By definition, G1(M ) and Gµ1(M ) are C1-open subsets of X1(M ) and X1µ(M ), re-spectively.
Given that Definition 1.1 concerns only to critical points and that the hyperbolicity on critical points is merely orbit-wise, the star property looks, a priori, quite a weak
X1 µ(M ) G1 µ(M ) X1(M ) G1(M )
Figure 1.3: Representation of the sets G1(M ) and Gµ1(M ).
property. However, as stated in Theorem 1 ahead, for the divergence-free setting, it is not.
Let us now state the usual definition of uniformly hyperbolic set.
Definition 1.2 Given X ∈ X1(M ), an Xt-invariant, compact and regular set Λ on M is
uniformly hyperbolic if there exist a DXt-invariant splitting T
ΛM = EΛs ⊕ RX(Λ) ⊕ EΛu
and constants c > 0 and 0 < κ < 1 such that, for any x ∈ Λ and any t > 0, we have: DXt x|Es x ≤ cκt and DX −t Xt(x)|Eu Xt(x) ≤ cκ t,
where RX(x) denotes the space spanned by Xt(x).
Observe that the constants c and κ, in the previous definition, do not depend on x ∈ Λ. The definition of Anosov vector field is related with the definition of uniformly hy-perbolic set.
Definition 1.3 A C1-vector field X defined on M is called Anosov if the manifold M is
uniformly hyperbolic. Let A1(M ) denote the set of Anosov C1−vector fields and denote
by A1
µ(M ) the set of Anosov C1-divergence-free vector fields defined on M .
The sets A1(M ) and A1
µ(M ) are C1-open subsets of X1(M ) and X1µ(M ), respectively
(see [5]).
Remark 1 Note that, if X is an Anosov vector field then Sing(X) = ∅. In fact, if there is q ∈ Sing(X) then q is hyperbolic, therefore isolated and satisfying TqM = Eqs⊕ Equ.
This means that q is surrounded by regular hyperbolic points p satisfying TpM = Eps⊕ RX(p) ⊕ E
u p.
But this is a contradiction, since the fibers of TxM depend continuously on x ∈ M .
A star vector field may fail to have a hyperbolic non-wandering set, as the famous Lorenz attractor shows (see [39]), since the hyperbolic saddle-type singularity is ac-cumulated by hyperbolic closed orbits, which are contained in the non-wandering set. This prevents the flow to be Axiom A. There are also examples of star vector fields that fail to have the critical elements dense in the non-wandering set (see [31]) or, even satisfying the Axiom A property, still may fail to satisfy the no-cycle condition (see [49]). However, all these star vector fields counterexamples exhibit singularities. Recently, Gan and Wen proved, in [37], that a star C1-vector field defined on a
d-dimensional manifold (d ≥ 3) with no singularities is Axiom A without cycles. Later, based in lower-dimensional conservative-type seminal ideas of Ma˜n´e and on the open-ness of the set of Anosov divergence-free vector fields, Bessa and Rocha proved, in [20], that G1
µ(M3) = A1µ(M3). The proof of this result cannot be trivially adapted to higher
dimensions. We remark that, in dimension 3, divergence-free vector fields with a domi-nated splitting are, in fact, Anosov. This happens because the normal bundle is splitted in two 1-dimensional subbundles (see [20, Lemma 3.2]). However, this is not necessarily true in higher dimensions.
The first theorem is the high-dimensional version of this later result and it is used to derive the proof of Conjecture 1.2 in the C1-divergence-free vector fields context.
Theorem 1 ([34, Theorem 1]) If X ∈ G1
µ(Md) then Sing(X) = ∅ and X ∈ A1µ(Md),
for d ≥ 4.
The main novelties in the proof of Theorem 1 are the use of a new strategy to prove the absence of singularities and the adaption of an argument of Ma˜n´e in [51] to show hyperbolicity from a dominated splitting, which follows easily when we are in dimension 3.
So, from the 3-dimensional result due to Bessa and Rocha and from Theorem 1, we have that Gµ1(Md) = A1µ(Md), for d ≥ 3.
The structural stability conjecture can also be stated in the Hamiltonian context. For that, we need to use specific tools and several recent results on Hamiltonian dynamics. It is worth pointing out that part of the difficulty of this problem consists in transposing in a proper way concepts from the general vector field setting to the Hamiltonian one.
Let (M2d, ω) be a symplectic manifold, where M2d (d ≥ 2) is an even-dimensional, compact, boundary-less, connected and smooth Riemannian manifold, endowed with a symplectic form ω. Denote by Cs(M, R) the set of Cs-real-valued functions on M and
call H ∈ Cs(M, R) a Cs-Hamiltonian, for s ≥ 2. From now on, we set s = 2.
H(p) R
p M
H
Figure 1.4: Representation of a Hamiltonian function H.
Given a Hamiltonian H, we can define the Hamiltonian vector field XH by
ω(XH(p), u) = dpH(u), ∀u ∈ TpM,
which generates the Hamiltonian flow XHt.
Remark 2 Observe that H is C2 if and only if XH is C1 and that, since H is continuous
and M is compact and boundary-less, Sing(XH) 6= ∅.
A scalar e ∈ H(M ) ⊂ R is called an energy of H. An energy hypersurface EH,e is a
connected component of H−1({e}), called energy level set.
R
e H
EH,e,4
EH,e,2 EH,e,3 EH,e,5
EH,e,1
The energy level set H−1({e}) is said regular if any energy hypersurface of H−1({e}) is regular. In this case, we can also say that the energy e is regular. Observe that a regular energy hypersurface is a XHt -invariant, compact and (2d − 1)-dimensional manifold.
Definition 1.4 Consider a Hamiltonian H ∈ C2(M, R), an energy e ∈ H(M) and a regular energy hypersurface EH,e. The triplet (H, e, EH,e) is called Hamiltonian system
and the pair (H, e) is called Hamiltonian level.
A Hamiltonian level (H, e) is said regular if the energy level set H−1({e}) is regular. If (H, e) is regular then H−1({e}) corresponds to the union of a finite number of closed connected components, that is, H−1({e}) = tIe
i=1EH,e,i, for Ie ∈ N. EH,e,1 . . H −1({e}) . . EH,e,Ie
Figure 1.6: Representation of a regular energy level.
Fixing a small neighborhood W of a regular energy hypersurface EH,e, there exist a
small neighborhood U of the Hamiltonian H and > 0 such that, for any ˜H ∈ U and for any ˜e ∈ (e − , e + ), we have ˜H−1({˜e}) ∩ W = EH,˜˜e. The energy hypersurface EH,˜˜ e
is called analytic continuation of EH,e.
EH,e
W E
˜ H,˜e
Figure 1.7: Representation of a analytic continuation of EH,e.
Accordingly with the previous notions, we introduce the definition of Hamiltonian star system.
Definition 1.5 A Hamiltonian system (H, e, EH,e) is called a Hamiltonian star system
˜
e ∈ (e − , e + ), all the closed orbits of EH,˜˜e are hyperbolic. We denote by EH,e?
the regular energy hypersurface with the previous property and by Gω2(M2d) the set of triplets of all Hamiltonian star systems defined on a 2d-dimensional symplectic manifold, for d ≥ 2.
Note that a Hamiltonian H can appear several times in the triplets in G2
ω(M2d). This
is possible if H is followed by a different energy e or, even with the same energy, if it is grouped with a different energy hypersurface.
The next definition states when a Hamiltonian system is Anosov.
Definition 1.6 A Hamiltonian system (H, e, EH,e) is Anosov if EH,e is uniformly
hyper-bolic for the Hamiltonian flow XHt associated to H. Let A2ω(M2d) denote the set of triplets of Anosov Hamiltonian systems, defined on a 2d-dimensional symplectic mani-fold, for d ≥ 2.
To prove Conjecture 1.1 in the Hamiltonian context, we need to prove that the set of Anosov Hamiltonian systems is open and that its elements are structurally stable. For such, let us state the definition of an open set of Hamiltonian systems and of a structurally stable Hamiltonian system.
Definition 1.7 Let H be a set of Hamiltonian systems. The set H is open if, for any Hamiltonian system (H, e, EH,e) ∈ H, there exist a small neighborhood U of H and
> 0 such that, for any ˜H ∈ U and any ˜e ∈ (e − , e + ), the Hamiltonian system ( ˜H, ˜e, EH,˜˜ e) belongs to H.
Note that the neighborhood of (H, e, EH,e) ∈ H is determined by U and .
The following result, refers to the openness of Anosov Hamiltonian systems defined on a 2d-dimensional symplectic manifold (d ≥ 2).
Theorem 2 ([13, Theorem 3]) The set A2
ω(M2d) is open, for d ≥ 2.
Definition 1.8 Consider a Hamiltonian system (H, e, EH,e). If there exist a small C2
-neighborhood U of H and > 0 such that, for any ˜H ∈ U and any ˜e ∈ (e−, e+) there exists a homeomorphism between EH,e and EH,˜˜ e, preserving orbits and their orientations,
we say that the Hamiltonian system (H, e, EH,e) is C2-structurally stable.
From this definition, we have the following result.
Theorem 3 ([13, Theorem 3]) The elements of A2
ω(M2d) are C2-structurally stable,
for d ≥ 2.
Now, we are in conditions to state the version of Conjecture 1.2 for Hamiltonians.
Theorem 4 ([13, Theorem 1]) If (H, e, EH,e? ) ∈ Gω2(M4) then (H, e, EH,e? ) ∈ A2ω(M4).
The previous theorem states that a Hamiltonian star system, defined on a 4-dimen-sional symplectic manifold, is, in fact, an Anosov Hamiltonian system. To prove this, we follow the strategy described by Bessa and Rocha, in [20], for conservative flows. This result is only obtained in dimension 4 because its proof makes use of some results that are only available in low dimension.
From Theorem 1 and Theorem 4, we can derive some interesting results, as an answer to the structural stability conjecture. Let us start with the definition of structurally stable vector field.
Definition 1.9 A C1-vector field X is called C1-structurally stable if there exists a C1 -neighborhood U of X in X1(M ) such that, for any Y ∈ U , there exists a homeomorphism between Xtand Yt, preserving orbits and their orientations. Denote by SS1(M ) the set of C1-structurally stable vector fields and by SS1
µ(M ) the set of C1-structurally stable
divergence-free vector fields.
It is also well-known that Anosov C1-vector fields are C1-structurally stable (see [5]). Hence, Conjecture 1.1 states the equivalence between uniform hyperbolicity and C1-structural stability.
Theorem 5 ([34, Theorem 2]) If X ∈ SS1µ(Md) then X ∈ A1µ(Md), for d ≥ 4.
The following result is a 4-dimensional proof of the structural stability conjecture for Hamiltonians. It says that a C2-structurally stable Hamiltonian system, defined on a
4-dimensional symplectic manifold, is Anosov.
Theorem 6 ([13, Theorem 2]) If (H, e, EH,e) is a C2-structurally stable Hamiltonian
system then (H, e, EH,e) ∈ A2ω(M4).
Now, we want to state some other consequences of Theorem 1 and Theorem 4. For such, we introduce some extra definitions.
Definition 1.10 Let V be an open subset of X1
µ(M ). We say that a C1-vector field X
is isolated in the boundary of the set V if X /∈ V and, given a small neighborhood U of X, any vector field Y ∈ U \X belongs to V.
U V Y X X1 µ(M )
Figure 1.8: Vector field X isolated in the boundary of a set V.
Accordingly with this definition, by Theorem 1, we obtain the following result.
Corollary 1 ([34, Corollary 1]) The boundary of the set A1
µ(Md) has no isolated points,
for d ≥ 4.
We can also try to describe the boundary of a Hamiltonian system.
Definition 1.11 Let H be a set of Hamiltonian systems. We say that a Hamilto-nian system (H, e, EH,e) is isolated in the boundary of H if (H, e, EH,e) /∈ H but,
given any small C2-neighborhood U of H and δ > 0, for any ˜H ∈ U \H and for any ˜
Now, following Theorem 4, we can derive an analogous result to Corollary 1, but for 4-dimensional symplectic manifolds.
Corollary 2 ([13, Corollary 1]) The boundary of the set A2
ω(M4) has no isolated points.
Now, we want to state a corollary of Theorem 1, concerning on Kupka-Smale vector fields.
Definition 1.12 A vector field X ∈ X1(M ) is Kupka-Smale if all the elements of the set
Crit(X) are hyperbolic and their stable and unstable manifolds intersect transversely. Denote by KS1(M ) the set of C1-Kupka-Smale vector fields and by KS1µ(M ) the set of C1-Kupka-Smale divergence-free vector fields.
Wu X(p)
Ws X(p)
p
Figure 1.9: Representation of a critical point p of a Kupka-Smale vector field.
See Section 2.1.3, for more details on the invariant manifolds of a hyperbolic set. In [73], Smale shows that the set KS1(M ) is a C1-residual subset of X1(M ). Later, Robinson proved this property for divergence-free vector fields. So, the set KS1µ(M ) is a C1-residual subset of X1
µ(M ) (see [69]). From [20, Theorem 1.2] and Theorem 1, it
is straightforward to obtain the following result.
Corollary 3 If X ∈ int(KS1µ(Md)) then X ∈ A1µ(Md), for d ≥ 3.
We remark that int(S) stands for the C1-interior of the set S ⊂ X1
µ(M ). This means
that Theorem 1 gives an immediate proof, for divergence-free vector fields, of the result shown by Toyoshiba, in [75].
In this section, we have emphasized the implication of Theorem 1 and Theorem 4 in the proof of the structural stability conjecture for high-dimensional divergence-free
vector fields and for 4-dimensional Hamiltonian systems. It was also stated that these theorems lead to some other results.
1.2
Shadowing and expansiveness
The theory of shadowing studies the closeness of pseudo-orbits and exact trajectories of dynamical systems. A dynamical system has some shadowing property if any pseudo-orbit with small error is, in some sense, close to some exact trajectory. The notions of pseudo-orbit and being close can be formalized in several ways. Therefore, since Anosov and Bowen various types of shadowing properties have been introduced in several contexts.
We want to state the definition of shadowing for continuous-time systems. First, de-fine Rep as the set of the increasing homeomorphisms α : R → R, called reparametriza-tions, satisfying α(0) = 0. Fixing > 0, define the set
Rep() = n α ∈ Rep : α(t) t − 1 < , t ∈ R\{0} o .
When we choose a reparametrization α in the previous set, we want α(t) to be taken arbitrarily close to the identity.
Definition 1.13 Fix T > 0 and δ > 0. A map ψ : R → M is a (δ, T )-pseudo-orbit of a flow Xt if dist(Xt(ψ(τ )), ψ(τ + t)) < δ, for any τ ∈ R and any |t| ≤ T . A pseudo-orbit
ψ of a flow Xt is said to be -shadowed by some orbit of Xt if there is x ∈ M and a
reparametrization α ∈ Rep() such that dist(Xα(t)(x), ψ(t)) < , for every t ∈ R.
ψ(τ1) XT(ψ(τ 1)) ψ(τ1+ T ) X−T(ψ(τ1)) ψ(τ1− T ) ψ(τ2+ T ) XT(ψ(τ 2)) ψ(τ2) X−T(ψ(τ 2)) ψ(τ2− T )
Figure 1.10: Representation of a pseudo-orbit.
Note that ψ is not assumed to be continuous.
Now, we are ready to properly state the definition of shadowing for C1-vector fields, in which we need a reparameterization of shadowing orbits.
Definition 1.14 A C1-vector field X satisfies the shadowing property if, for any > 0 and any T > 0, there is δ > 0 such that any (δ, T )-pseudo-orbit ψ is -shadowed by some orbit of X. Let S1(M ) and Sµ1(M ) denote the sets of vector fields in X1(M ) and X1
µ(M ), respectively, satisfying the shadowing property.
Smale proved that a diffeomorphism in the C1-interior of the set of diffeomorphisms
with the shadowing property is C1-structurally stable (see [73]). More recently, Lee and
Sakai proved, in [47], that if X belongs to the interior of the set S1(M ) and has no
singularities then X satisfies the Axiom A and the strong transversality conditions. For divergence-free vector fields, we prove the following result.
Theorem 7 ([33, Theorem 1]) If X ∈ int(Sµ1(Md)) then X ∈ A1µ(Md), for d ≥ 3.
The Lipschitz shadowing property is a stronger definition of shadowing.
Definition 1.15 A C1-vector field X satisfies the Lipschitz shadowing property if there are positive constants ` and δ0 such that any (δ, T )-pseudo-orbit ψ, with T > 0 and
δ ≤ δ0, is `δ-shadowed by an orbit of X. Let LS1(M ) and LS1µ(M ) denote the sets
of vector fields in X1(M ) and X1
µ(M ), respectively, satisfying the Lipschitz shadowing
property.
By definition, it is immediate that the set LS1(M ) is a subset of S1(M ) and that
the set LS1µ(M ) is a subset of Sµ1(M ). Therefore, from Theorem 7, we have that the C1-interior of the set LS1µ(M ) is contained in the set A1µ(M ).
In [74], Tikhomirov proved that a vector field in the C1-interior of the set of vector fields with the Lipschitz shadowing property is structurally stable. Recently, Pilyugin and Tikhomirov proved that a C1-diffeomorphism having the Lipschitz shadowing property
is structurally stable (see [64]).
The following definition is the notion of expansive vector field, introduced by Bowen and Walters, in [28].
Definition 1.16 A C1-vector field X is expansive if, for any > 0, there is δ > 0 such that if x, y ∈ M satisfy dist(Xt(x), Xα(t)(y)) ≤ δ, for any t ∈ R and for some continuous map α : R → R with α(0) = 0, then y = Xs(x), where |s| ≤ . Denote by
E1(M ) ⊂ X1(M ) the set of expansive vector fields and by E1
µ(M ) ⊂ X1µ(M ) the set of
expansive divergence-free vector fields, both endowed with the C1 Whitney topology.
x
y = Xs(x)
Xt(x)
Xα(t)(y)
Figure 1.11: Representation of an expansive vector field’s orbit.
This definition asserts that any two points whose orbits remain indistinguishable, up to any continuous time displacement, must be in the same orbit.
Observe that the reparametrization α, in Definition 1.16, is not assumed to be close to identity and that the expansiveness property does not depend on the choice of the metric on M .
In 1970’s, Ma˜n´e proved that a diffeomorphism f in the C1-interior of the set of expansive diffeomorphisms is Axiom A and satisfies the quasi-transversality condition (see [53]). Later, Moriyasu, Sakai and Sun proved the same result for vector fields, in [57]. Moreover, the authors proved that if X ∈ int(E1(M )) and has the shadowing
property then X is Anosov. Recently, Pilyugin and Tikhomirov proved that an expansive diffeomorphism having the Lipschitz shadowing property is Anosov (see [64]). In the next result, we prove that a divergence-free vector field in the C1-interior of the set of
expansive divergence-free vector fields is actually Anosov.
Theorem 8 ([33, Theorem 1]) If X ∈ int(E1
µ(Md)) then X ∈ A1µ(Md), for d ≥ 3.
The expansiveness and the shadowing properties play an essential role in the inves-tigation of the stability theory and the ergodic theory of Axiom A diffeomorphisms (see [26]). It is well-known that Anosov systems are expansive and satisfy the shadowing and the Lipschitz shadowing properties (see [5, 63]).
To conclude this section, we notice that, by Theorem 1, Theorem 7, Theorem 8 and Corollary 3, we have the following result.
Corollary 1.1 For the conservative setting, G1 µ(M ) = A1µ(M ) = int(Sµ1(M )) = int(KS 1 µ(M )) = int(LS 1 µ(M )) = int(Eµ1(M )).
1.3
General scenario for dynamics
At the second half of the 1960’s, it was already clear that the set of uniformly hyperbolic systems is open but not dense. Thus, it triggered the beginning of the search for an answer to the following question.
Question 1.1 Is it possible to look for a general scenario for dynamics?
This search draws the attention to homoclinic orbits, that is, orbits that in the past and in the future converge to the same periodic orbit, which have been firstly considered by Poincar´e, almost a century before. The creation or destruction of such orbits is, roughly speaking, what its meant by homoclinic bifurcations (see, for example, [62]). Based on these and other subsequent developments, in [60], Palis formulated Conjecture 1.3, con-cerning on hyperbolicity, homoclinic tangencies and heterodimensional cycles. Roughly speaking, a homoclinic tangency is a non-transverse intersection between the stable and unstable manifolds of a hyperbolic closed orbit of saddle-type. A heterodimensional cycle is a cyclical intersection between the invariant manifolds of two distinct hyper-bolic critical points of saddle-type with different dimension of the unstable bundles (see Definition 2.6, in Section 2.1.3, for more details).
Conjecture 1.3 Diffeomorphisms with either a homoclinic tangency or a heterodimen-sional cycle are Cr-dense in the complement of the Cr closure of hyperbolic
diffeomor-phisms (r ≥ 1).
In [67], Pujals and Sambarino proved this conjecture in the case of C1
-diffeomorph-isms defined on a compact surface. Recently, Bessa and Rocha proved this conjecture for C1-preserving diffeomorphisms in [16]. In fact, the authors show that a
volume-preserving diffeomorphism can be C1-approximated by an Anosov volume-preserving
diffeomorphism, or else by a volume-preserving diffeomorphism displaying a heterodi-mensional cycle. The authors also proved a similar result for symplectomorphisms.
For the continuous-time case, Arroyo and Hertz proved, in [9], an analogous state-ment of Conjecture 1.3 for C1-vector fields defined on a 3-dimensional, compact man-ifold. In this context, besides homoclinic tangencies and heterodimensional cycles, the
singular cycles are another homoclinic phenomenon that must be considered. The au-thors show that a vector field X ∈ X1(M3) can be C1-approximated by an Anosov vector field, or else by a vector field displaying a homoclinic tangency, or else by a vector field displaying a singular cycle. For the divergence-free context, Bessa and Rocha show, in [18], that any vector field X in X1
µ(M3) can be C1-approximated by a divergence-free
vector field which is Anosov, or else has a homoclinic tangency. In this paper, the authors left open the following question, related with Conjecture 1.3.
Question 1.2 Can any vector field X in X1
µ(Md) be C1-approximated by a
divergence-free vector field exhibiting some form of hyperbolicity on Md(d ≥ 4), or by one exhibiting homoclinic tangencies, or else by one having a heterodimensional cycle?
The following result is the answer to this question.
Theorem 9 ([34, Theorem 3]) If X ∈ X1
µ(Md), for d ≥ 4, then X can be C1
-approximated by an Anosov divergence-free vector field, or else by a divergence-free vector field exhibiting a heterodimensional cycle.
1.4
Topological transitivity
The topological transitivity is a global property of a dynamical system. As a moti-vation for this notion, we may think of a real physical system, where a state is never measured exactly. Thus, instead of points, we should study (small) open subsets of the phase space and describe how they move in that space. If each one of these open subsets meet each other by the action of the system after some time, then we say that the system is topologically transitive. Equivalently, if we take a compact phase space, we may say that the system has a dense orbit. However, if the open subsets remain inseparable after some time, by the iteration of the system, then we say that the system is topologically mixing. Obviously, a topologically mixing system is also a topologically transitive system.
The concept of transitivity goes back to Birkhoff. According to [38], Birkhoff used it in [21, 22]. Throughout in this thesis transitive will always mean topologically transitive.
There exists a lot of transitive systems, as the irrational rotations of S1, the shift maps and the basic sets. It is also well-known that C1+α-Anosov systems are ergodic and so transitive (see [5]). Nevertheless, transitivity is not an open property.
Question 1.3 Can the transitivity property be generic?
Some authors have been working on this question. The first remarkable result on this subject is due to Bonatti and Crovisier, in [24]. The authors show that, C1-generically, a C1-conservative diffeomorphism is transitive. Later, jointly with Arnaud, the authors extend this result for C1-symplectic diffeomorphisms defined on a symplectic manifold (see [8]). Adapting the techniques used to prove these results to the continuous-time case, Bessa proved an analogous result for C1-divergence-free vector fields. In fact,
by a result due to Abdenur, Avila and Bochi (see [1]), Bessa was able to show that, C1-generically, a divergence-free vector field is topologically mixing (see [11]).
Our contribution to this issue is the statement and the proof of a result that is an answer to Question 1.3 for Hamiltonian systems. Let us start with some definitions.
Definition 1.17 A compact energy hypersurface EH,e is topologically mixing if, for
any open and non-empty subsets of EH,e, say U and V , there is τ ∈ R such that
XHt(U ) ∩ V 6= ∅, for any t ≥ τ . A regular Hamiltonian level (H, e) is topologically mixing if each one of the energy hypersurfaces of H−1({e}) is topologically mixing.
Accordingly with this definition, we prove the following result.
Theorem 10 There exists a residual set R in C2(M, R) such that, for any H ∈ R,
there is an open and dense set S(H) in H(M ) such that, for every e ∈ S(H), the Hamiltonian level (H, e) is topologically mixing.
The main tool to prove the previous result is a version for Hamiltonians of the Connecting Lemma for pseudo-orbits developed in [8] by Arnaud, Bonatti and Crovisier. To state it, we need the notions of resonance relations and of pseudo-orbits, which we postpone to Section 3.1.5 and Section 3.1.6.
Lemma 1 (Connecting Lemma for pseudo-orbits of Hamiltonians) Let (M, ω) denote a compact, symplectic 2d-manifold, for d ≥ 2. Take H ∈ C2(M, R) and a regular energy e ∈ H(M ), such that the eigenvalues of any closed orbit of H do not satisfy non-trivial resonances. Then, for any C2-neighborhood U of H, for any energy
hypersurface EH,e ⊂ H−1({e}) and for any x, y ∈ EH,e connected by an -pseudo-orbit,
for > 0, there exist ˜H ∈ U and t > 0 such that e = ˜H(x) and Xt ˜
H(x) = y on the
analytic continuation EH,e˜ of EH,e.
x y Xt ˜ H(x) EH,e EH,e˜
Figure 1.12: Representation of the analytic continuation of EH,e.
To prove these results, we have to resume the arguments used by Arnaud, Bonatti, Crovisier and Bessa in [8, 11, 24] and to adapt it to the Hamiltonian setting. The main change in the proofs is the need to restrict attention to the energy hypersurface, when analyzing the perturbations and their supports.
From Theorem 10, we can derive the following result concerning on the homoclinic class of a hyperbolic closed orbit γ of H, which is the closure of the set of transver-sal intersections between the stable and unstable manifolds of all points p in γ (see Section 3.1.4, for more details).
Corollary 4 There is a residual set R in C2(M, R) such that, for any H ∈ R, there is an
open and dense set S(H) in H(M ) such that if e ∈ S(H) then any energy hypersurface of H−1({e}) is a homoclinic class.
If any energy hypersurface of H−1({e}) is a homoclinic class, we say that H−1({e}) is a homoclinic class.
We end this chapter with an overview of the remaining chapters of this thesis. This thesis is organized in four additional chapters. In Chapter 2, we include the proofs of the results on conservative dynamics and in Chapter 3 we concern about the proofs of the results on Hamiltonian dynamics. In each chapter we also include extra definitions and useful auxiliary results. In the last chapters, we synthesize the main results of this thesis and we describe some ideas to improve and to develop this work.
TWO
CONSERVATIVE DYNAMICS
This chapter begins with some extra definitions on conservative dynamics and it includes the statement of some auxiliary results. After, Section 2.2 brings together the complete proofs of the results on conservative dynamics, that is, of Theorem 1, Theorem 5, Theorem 7, Theorem 8, Theorem 9 and Corollary 1.
2.1
Definitions and auxiliary results
In this section, we state the definition of Lyapunov exponents, of the Linear Poincar´e flow and of heterodimensional cycles. Afterwards, we state some perturbation results that will be used to complete the proofs, in Section 2.2.
2.1.1
Lyapunov exponents and classification of closed orbits
This section is about Lyapunov exponents for the conservative continuous-time case and their properties. Firstly, we remark that the Riemannian structure on M induces a norm k.k on the fibers TpM , ∀ p ∈ M . From now on, we use the standard norm of a
bounded linear map L : T M → T M given by kLk = sup
kuk=1
kL(u)k .
Given X ∈ X1µ(M ), Oseledets’ theorem (see [59]) ensures that µ-almost every point x ∈ M admits a splitting of the tangent bundle,
TxM = Ex1⊕ · · · ⊕ E k(x) x
and also real numbers λ1(x) > · · · > λk(x)(x), for 1 ≤ k(x) ≤ d, called Lyapunov
exponents, such that DXxt(Exi) = EXi t(x) and
λi(x) = lim t→±∞ 1 t log kDX t x(v i)k, for any vi ∈ Ei
x\ {~0} and i ∈ {1, ..., k(x)}. This splitting is called Oseledets’ splitting.
The full µ-measure set of Oseledets’ points is denoted by O(X). Remark 3 As a consequence of Oseledets’s theorem, we have that
k(x) X i=1 λi(x) · dim(Exi) = limt→±∞ 1 t log | det DX t x|.
However, since the vector field X is divergence-free, we deduce that | det DXt(x)| = 1,
for any t ∈ R and any x ∈ M . Therefore, we conclude that
k(x)
X
i=1
λi(x) · dim(Exi) = 0, ∀ x ∈ O(X).
Note that if we do not take into account the multiplicities of the eigenvalues associated to the eigenspaces E1
x, · · ·, E k(x)
x , we have exactly d = dim(M ) Lyapunov exponents,
λ1(x) ≥ · · · ≥ λd(x).
Let γ ⊂ M be a closed orbit of period π and fix p ∈ γ. The characteristic multipliers of γ are the eigenvalues of DXpπ, which are independent of p ∈ γ. If σ is a characteristic multiplier of γ, then the associated Lyapunov exponent is λ = log(σ)/π. A characteristic multiplier σ is said simple if its multiplicity is equal to 1.
Definition 2.1 A closed orbit γ ⊂ M is called
• hyperbolic, when all the characteristic multipliers have modulus different from 1; • parabolic, when at least one of the characteristic multipliers is real and of modulus
1;
• completely elliptic, when all the characteristic multipliers are simple, non-real and of modulus 1;
• elliptic, when γ has at least two simple, non-real and of modulus 1 characteristic multipliers.
Figure 2.1: Representation of the spectrum of a hyperbolic, a parabolic, a completely elliptic and an elliptic closed orbit, respectivelly.
Notice that, given an elliptic or a completely elliptic closed orbit γ, if we do not assume the characteristic multipliers of γ to be simple then, under small perturbations, we are able to turn γ into a hyperbolic closed orbit. The same happens if we take a parabolic orbit. (1) (1) (1) (1) (2) (2) perturbation
Figure 2.2: Transformation of a completely elliptic closed orbit, with no simple characteristic multipliers, into a hyperbolic closed orbit.
2.1.2
Linear Poincar´
e flow and hyperbolicity
In this section, we define the linear Poincar´e flow and we state some results related with this flow. Let us start with some definitions.
Given X in X1(M ) and a regular point x in M , let Nx := X(x)⊥ ⊂ TxM
denote the (dim(M ) − 1)–dimensional normal bundle of X at x and define Nx,r := Nx ∩ {u ∈ TxM : kuk < r}, for r > 0. Note that, in general, Nx is not
DXt
x-invariant.
Definition 2.2 The flow PXt(x) := ΠXt(x)◦ DXxt is called linear Poincar´e flow, where
ΠXt(x) : TXt(x)M → NXt(x) is the canonical orthogonal projection.
Recently, Li, Gan and Wen generalized the notion of the linear Poincar´e flow, in order to include singularities (see [48]).
NXt(p) Xt(p) Np v p X(Xt(p)) DXt p(v) Pt X(p)v X(p)
Figure 2.3: Representation of the linear Poincar´e flow.
Lemma 2.1 ([56, Lemma 3.10]) Consider X ∈ X1(M ) and Λ ⊂ M a compact, Xt
-invariant, regular set and assume that EΛ = EΛ1 ⊕ EΛ2. If there exists T > 0 such that
DXxT|E1 x ≤ 1/2 and DX −T XT(x)|E2 XT (x)
≤ 1/2, for every x ∈ Λ, then there are c > 0 and 0 < κ < 1 such that DXt
x|E1 x < cκt and DX −t Xt(x)|E2 Xt(x) < cκ t, for every x ∈ Λ and t > 0.
Taking into account the previous lemma, we state the following definition of uniformly hyperbolic set by using the linear Poincar´e flow.
Definition 2.3 Fix X ∈ X1(M ). An Xt-invariant, compact and regular set Λ ⊂ M is uniformly hyperbolic if NΛ admits a PXt-invariant splitting NΛs ⊕ NΛu such that there is
` > 0 satisfying kP` X(x)|Ns xk ≤ 1 2 and kP −` X (X`(x))|Nu X`(x)k ≤ 1 2, for any x ∈ Λ.
Observe that the constant 12 can be replaced by any constant θ ∈ (0, 1). If θ is close to 1, we say that the hyperbolicity is weak.
Supported on an abstract invariant manifold theory result of Hirsch, Pugh and Shub (see [44, Lemma 2.18]), in [32] Doering proves that the definition of uniformly hyperbolic compact set by using the linear Poincar´e flow (Definition 2.3) is equivalent to the usual definition of uniformly hyperbolic set of a flow (see Definition 1.2).
Lemma 2.2 ([32, Proposition 1.1]) Let Λ be a Xt-invariant, regular and compact set.
Then Λ is uniformly hyperbolic for Xt if and only if Λ is uniformly hyperbolic for Pt X.
It is straightforward to see that the definition of Lyapunov exponent, stated in Sec-tion 2.1.1, can also be adapted in order to use PXt instead of DXt. Hence, µ-almost
every point x ∈ M admits the Oseledets splitting Nx = Nx1⊕ · · · ⊕ N
k(x) x ,
for any 1 ≤ k(x) ≤ dim(M ) − 1, and the Lyapunov exponent
λi(x) = lim t→±∞ 1 t log kP t X(x)v ik, for any vi ∈ Ni x\ {~0} and i ∈ {1, ..., k(x)}.
A singularity p of a C1-vector field X is hyperbolic if the eigenvalues of DX p are
not purely imaginary. In the divergence-free context, a hyperbolic critical point p must be of saddle-type. If p is a closed orbit then the dimension of the fibers Ns
p and Npu is
between 1 and dim(M ) − 2.
Now, we state the definition of dominated splitting, which is weaker that the defini-tion of uniform hyperbolicity. For this, we use the linear Poincar´e flow.
Definition 2.4 Let X ∈ X1(M ) and let Λ ⊂ M be a compact, Xt-invariant and regular
set. Assume that there exists a Pt
X-invariant splitting N = N1 ⊕ · · · ⊕ Nk over Λ, for
1 ≤ k ≤ dim(M ) − 1, such that all the subbundles have constant dimension. This splitting is dominated if there exists ` > 0 such that, for any 0 ≤ i < j ≤ k,
kP` X(x)|Ni xk · kP −` X (X `(x))| Nj X`(x) k ≤ 1 2, for any x ∈ Λ.
Note that a vector field with a dominated splitting structure is not necessarily uni-formly hyperbolic.
Let us briefly state some useful properties of a dominated splitting over a set Λ (see [25] for more details):
• Uniqueness: the dominated splitting is unique, if the dimension of the subbundles is fixed.
• Continuity : any dominated splitting is continuous, that is, the subbundles N1 x and
N2
x depend continuously on the point x ∈ Λ.
• Extension to the closure: any `-dominated splitting over a set Λ can be extended to an `-dominated splitting over the closure of Λ.
• Extension to a neighborhood : the dominated splitting can be extended to the maximal flow-invariant set in a neighborhood of Λ.
• Persistence: any dominated splitting persists under C1-perturbations.
Remark 4 If we assume that there is not a dominated splitting on a flow-invariant, compact and regular set, it is possible to make a small C1-perturbation on the vector field in order to get a new one with Lyapunov exponents arbitrarily close to zero, as it is shown by Bessa and Rocha in [17, Theorem 1].
The next result corresponds to a dichotomy for C1-divergence-free vector fields.
It requires the existence of a closed orbit with arbitrarily large period. The proof of Theorem 2.1 for divergence-free vector fields follows the ideas stated in the proof of [19, Proposition 2.4].
Theorem 2.1 Let X ∈ X1
µ(M ) and let U be a small C1-neighborhood of X. Then, for
any > 0, there exist l > 0 and τ > 0 such that, for any Y ∈ U and any x ∈ P erτ(Y ),
• either Pt
Y admits an l-dominated splitting over the Yt-orbit of x, or else
• for any neighborhood U of x, there exists an -C1-perturbation ˜Y of Y , coinciding
with Y outside U and along the orbit of x, such that P˜π(x)
Y (x) has only eigenvalues
equal to 1 or −1, where π(x) stands for the period of x.
The following result says that if the vector field has a linear hyperbolic singularity of saddle-type then the linear Poincar´e flow cannot admit a dominated splitting over the set of regular points of M Note that a singularity p is linear if there exist smooth local coordinates around p such that X is linear and equal to DXp in these coordinates (see
[77, Definition 4.1]).
Proposition 2.1 [77, Proposition 4.1] If X ∈ X1(M ) has a linear hyperbolic singularity of saddle-type then PXt does not admit any dominated splitting over M \Sing(X).
We remark that the proof of this proposition can be easily adapted to the conservative case. Hence, Proposition 2.1 remains valid for C1-divergence-free vector fields.
We end this section with a lemma stating that a singularity can be turned into a linear one, by performing a small perturbation of the vector field.
Lemma 2.3 [19, Lemma 3.3] Let p be a singularity of X ∈ X1µ(M ). For any > 0, there exists Y ∈ X∞µ(M ) such that Y is -C1-close to X and p is a linear hyperbolic singularity of Y .
2.1.3
Heterodimensional cycles
This section contains the definition of heterodimensional cycle, as well as some useful remarks.
Consider a C1-vector field X and p ∈ Crit(X). Denote by O
X(p) the Xt-orbit of
p. We remark that if p is a singularity of X then we set OX(p) = p.
Definition 2.5 Let X be a C1-vector field and choose p in M . If O
X(p) is a hyperbolic
set, its stable and unstable manifolds are defined as follows:
WXs(OX(p)) = {q ∈ M : lim t→+∞dist(X t(q), O X(p)) = 0} and WXu(OX(p)) = {q ∈ M : lim t→+∞dist(X −t (q), OX(p)) = 0}.
We observe that both WXs(OX(p)) and WXu(OX(p)) do not depend on q ∈ OX(p).
Therefore, we can write Ws
X(OX(p)) = WXs(q) and WXu(OX(p)) = WXu(q), for some
q ∈ OX(p). These manifolds are respectively tangent to the subspaces Eqs⊕ RX(q) and
RX(q) ⊕ Equ of TqM , for q ∈ OX(p). Observe that
dim(WXs(OX(p))) + dim(WXu(OX(p))) = dim(M ) + i,
where i = 0 if p ∈ Sing(X) and i = 1 if p ∈ P er(X).
If p ∈ Crit(X) is a hyperbolic saddle its index is defined as the dimension of the unstable bundle Wu
X(p) and it is denoted by ind(p).
Definition 2.6 Consider X ∈ X1(M ) and let p, q be two distinct hyperbolic critical points of saddle-type such that ind(p) < ind(q). A vector field X exhibits a heterodi-mensional cycle associated to p and q if the invariant manifolds of p and q intersect cyclically, that is Ws
X(p) >∩ WXu(q) 6= ∅ and WXu(p) ∩ WXs(q) 6= ∅, where >∩ denotes
a transversal intersection. Let HC1(M ) ⊂ X1(M ) and HC1
µ(M ) ⊂ X1µ(M ) denote the
sets whose elements exhibit heterodimensional cycles.
Wu X(p) WXs(p) p q Ws X(q) Wu X(q)
Figure 2.4: Representation of a heterodimensional cycle.
We observe that, for reasons of simplicity, the previous figure represents, in fact, a heterodimensional cycle for the discrete time case.
Remark 5 The condition ind(p) < ind(q), in Definition 2.6, ensures that the connec-tion WXs(p) >∩ Wu
X(q) is C1-persistent and that the connection WXu(p) ∩ WXs(q) does
not persist under C1-generic perturbations.
We observe that Definition 2.6 can be trivially extended to a finite number of hyper-bolic saddles.
The next definition contains a classification of heterodimensional cycles.
Definition 2.7 A heterodimensional cycle is called • periodic, if it is composed just by closed orbits; • singular, if it is composed just by singularities;
